On denoising modulo 1 samples of a function

Abstract

Consider an unknown smooth function $f:[0,1]\to \mathbb{R}$, and say we are given $n$ noisy$\mathrm{mod}1$ samples of $f$, i.e., $y_i=(f(x_i)+{\eta }_i)\mathrm{mod}1$ for $x_i\in [0,1]$, where ${\eta }_i$ denotes noise. Given the samples $(x_i,y_i{)}_{i=1}^n$ our goal is to recover smooth, robust estimates of the clean samples $f(x_i)\mathrm{mod}1$. We formulate a natural approach for solving this problem which works with representations of mod 1 values over the unit circle. This amounts to solving a quadratically constrained quadratic program (QCQP) with non-convex constraints involving points lying on the unit circle. Our proposed approach is based on solving its relaxation which is a trust-region sub-problem, and hence solvable efficiently. We demonstrate its robustness to noise % of our approach via extensive simulations on several synthetic examples, and provide a detailed theoretical analysis.

19 pages, 13 figures. To appear in AISTATS 2018. Corrected typos, and made minor stylistic changes throughout. Main results unchanged. Added section I (and Figure 13) in appendix

References in corpus (1)

• Manopt, a Matlab toolbox for optimization on manifolds